U.S. patent application number 17/599154 was filed with the patent office on 2022-05-19 for phase transforming cellular materials.
The applicant listed for this patent is Gordon JARROLD, Nilesh MANKAME, PURDUE RESEARCH FOUNDATION, David RESTREPO, Maria Mirian VELAY-LIZANCOS, Pablo ZAVATTIERI, Yunlan ZHANG. Invention is credited to Gordon Jarrold, Nilesh Mankame, David Restrepo, Maria Milian Velay-Lizancos, Pablo Zavattieri, Yunlan Zhang.
Application Number | 20220154702 17/599154 |
Document ID | / |
Family ID | 1000006177199 |
Filed Date | 2022-05-19 |
United States Patent
Application |
20220154702 |
Kind Code |
A1 |
Zavattieri; Pablo ; et
al. |
May 19, 2022 |
PHASE TRANSFORMING CELLULAR MATERIALS
Abstract
A phase transformational cellular material, including a
plurality of bistable cells, each respective bistable cell
operationally connected to at least one other respective bistable
cell. Each bistable cell enjoys a first stable phase and a second
stable phase. The first stable phase is a first geometric
configuration and the second stable phase is a second geometric
configuration different from the first geometric configuration. An
energy transaction is required to shift each respective cell
between stable phases. A mechanical energy transaction is required
to shift from the first to the second phase, while a thermal energy
transaction is required to shift from the second to the first
phase.
Inventors: |
Zavattieri; Pablo; (West
Lafayette, IN) ; Zhang; Yunlan; (Foothill Ranch,
CA) ; Mankame; Nilesh; (Ann Arbor, MI) ;
Restrepo; David; (San Antonio, TX) ; Jarrold;
Gordon; (Chicago, IL) ; Velay-Lizancos; Maria
Milian; (West Lafayette, IN) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
ZAVATTIERI; Pablo
ZHANG; Yunlan
MANKAME; Nilesh
RESTREPO; David
JARROLD; Gordon
VELAY-LIZANCOS; Maria Mirian
PURDUE RESEARCH FOUNDATION |
West Lafayette
Foothill Ranch
Ann Arbor
San Antonio
Chicago
West Lafayette
West Lafayette |
IN
CA
MI
TX
IL
IN
IN |
US
US
US
US
US
US
US |
|
|
Family ID: |
1000006177199 |
Appl. No.: |
17/599154 |
Filed: |
March 27, 2020 |
PCT Filed: |
March 27, 2020 |
PCT NO: |
PCT/US2020/025197 |
371 Date: |
September 28, 2021 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
|
62826376 |
Mar 29, 2019 |
|
|
|
Current U.S.
Class: |
1/1 |
Current CPC
Class: |
F03G 7/06115 20210801;
E04H 9/0237 20200501; B60C 7/14 20130101 |
International
Class: |
F03G 7/06 20060101
F03G007/06; E04H 9/02 20060101 E04H009/02; B60C 7/14 20060101
B60C007/14 |
Goverment Interests
FUNDING STATEMENT
[0002] This invention was made with government support under
CMMI1538898 awarded by the National Science Foundation. The
government has certain rights in the invention.
Claims
1. A bistable multicellular body, comprising: a first phase
transforming cell; and a second phase transforming cell
operationally connected to the first phase transforming cell;
wherein each respective phase transforming cell may occupy a first
phase defined as a first stable geometry; wherein each respective
phase transforming cell may occupy a second phase defined as a
second, different, stable geometry; wherein a forward phase
transformation from the first phase to the second phase occurs in
response to an applied mechanical load.
2. The multicellular body of claim 1 wherein each phase
transforming cell includes a rigid portion and a flexible
portion.
3. The multicellular body of claim 2 wherein the rigid portion is a
pair of parallel rigid walls and the flexible portion is a pair of
parallel, curved beams extending between the pair of rigid
walls.
4. The multicellular body of claim 3 and further comprising a rigid
support wall extending between each pair of parallel curved
beams.
5. The multicellular body of claim 1 wherein a reverse phase
transformation from the second phase to the first phase occurs in
response to a thermal stimulus.
6. The multicellular body of claim 5 wherein the thermal stimulus
is an increase of thermal energy in the body.
7. The multicellular body of claim 5 wherein the body is bistable
at a first lower temperature and metastable at a second, higher
temperature.
8. The multicellular body of claim 1 wherein the multicellular body
further comprises a plurality of phase transforming cells
configured as a vehicle tire.
9. The multicellular body of claim 1 wherein the multicellular body
further comprises a plurality of phase transforming cells
configured as an earthquake-resistant building member.
10. The multicellular body of claim 1 and further comprising: a
plurality of phase transforming cells operationally connected to
define a hexagonal pattern; and a plurality of cylindrical shell
ligaments, each respective ligament operationally connected to two
phase transforming cells; wherein each respective phase
transforming cell exhibits hexagonal symmetry; and wherein each
respective phase transforming cell connects to six respective
cylindrical shell ligaments.
11. The multicellular body of claim 1 wherein each respective phase
transforming cell is characterized by two regimes displaying
positive stiffness and one regime displaying negative
stiffness.
12. A phase transformational cellular material; comprising: a
plurality of bistable cells, each respective bistable cell
operationally connected to at least one other respective bistable
cell; wherein each bistable cell enjoys a first stable phase and a
second stable phase; wherein the first stable phase is a first
geometric configuration; wherein the second stable phase is a
second geometric configuration different from the first geometric
configuration; wherein an energy transaction is required to shift
each respective cell between stable phases.
13. The phase transformational cellular material of claim 12,
wherein the plurality of bistable cells is configured into an
automobile tire.
14. The phase transformational cellular material of claim 13, and
further comprising a rubber shell encapsulating the plurality of
bistable cells.
15. The phase transformational cellular material of claim 13,
wherein the plurality of bistable cells have a square 2D PXCM
cross-sectional shape.
16. The phase transformational cellular material of claim 12,
wherein the plurality of bistable cells enjoys alternating regions
of marked wave propagation and wave attenuation; and wherein wave
propagation through the plurality of bistable cells enjoys strong
directionality.
17. The phase transformational cellular material of claim 12,
wherein the plurality of bistable cells is piezoresponsive.
18. The phase transformational cellular material of claim 12,
wherein the plurality of bistable cells are hydrophobic when in the
first stable phase; and wherein the plurality of bistable cells are
hydrophobic when in the second stable phase.
19. The phase transformational cellular material of claim 12,
wherein the a mechanical force is required to shift the plurality
of bistable cells from the first stable phase to the second stable
phase; and wherein a temperature increase is required to shift the
plurality of bistable cells from the second stable phase to the
first stable phase.
20. The phase transformational cellular material of claim 12
wherein the plurality of bistable cells are configured as an
earthquake-resistant structural member.
21. The phase transformational cellular material of claim 12
wherein a mechanical energy transaction is required to shift from
the first to the second phase; and wherein a thermal energy
transaction is required to shift from the second to the first
phase.
22. The phase transformational cellular material of claim 12
wherein the plurality of bistable cells are operationally connected
to define a hexagonal pattern; wherein each respective bistable
cell enjoys a hexagonal symmetry; wherein the phase
transformational cellular material further comprises a plurality of
cylindrical shell ligaments, each respective ligament operationally
connected to two bistable cells; and wherein each respective
bistable cell connects to six respective cylindrical shell
segments.
23. The multicellular body of claim 22 wherein each respective
bistable is characterized by two regimes displaying positive
stiffness and one regime displaying negative stiffness.
Description
CROSS-REFERENCE TO RELATED APPLICATIONS
[0001] This patent application claims priority to U.S. provisional
patent application Ser. No. 62/826,376, filed on Mar. 29, 2019.
TECHNICAL FIELD
[0003] The present novel technology relates generally to structural
or building materials, and, more specifically, to
phase-transformable cellular materials.
BACKGROUND
[0004] A phase transformation is the change of a thermodynamic
system from one phase to another. Martensitic phase transformations
play a fundamental role in the behavior of a large class of active
materials which include shape memory, ferroelectric and some
magnetostrictive alloys. Typical solid-state phase transformations
in materials result from a change in the packing arrangement of the
atoms in the unit cell. At the micro level, these changes can be
viewed as multi-stable devices that deform switching between
locally stable configurations, and macroscopically the switching
phenomena manifest through the evolution of the domain
microstructures in which the associated energy landscapes are
usually extremely wiggly. However, there remains a need to extend
this notion of solid-state phase transformations to cellular
materials, where phase transformations are represented by changes
in the geometry of its microstructure. The present novel technology
addresses this need.
BRIEF DESCRIPTION OF DRAWINGS
[0005] FIG. 1A is a schematic illustration of a phase transforming
unit cell of a first embodiment of the present novel
technology.
[0006] FIG. 1B schematically illustrates the force/displacement
relationship of a plurality of operationally connected phase
transforming cells of FIG. 1A.
[0007] FIG. 2 graphically illustrates the stress-strain
relationship of phase transforming cells.
[0008] FIG. 3A graphically illustrates the RMS velocity-frequency
relationship for a phase transforming metamaterial.
[0009] FIG. 3B illustrates normalized out-of-plane displacement
amplitudes for the metamaterial of FIG. 3A.
[0010] FIG. 4 schematically illustrates an energy harvesting device
utilizing phase transforming cellular technology of FIG. 1A.
[0011] FIG. 5A is a cross-sectional view of crushed tubes with and
without fillings of the metamaterial of FIG. 1A.
[0012] FIG. 5B graphically illustrates the stress-strain
relationship of the filled tube of FIG. 5A.
[0013] FIG. 6A illustrates a prior art fluid viscous damper.
[0014] FIG. 6B illustrates a prior art metallic damper.
[0015] FIG. 7A is a perspective view of a shape morphing tensegrity
structure.
[0016] FIG. 7B is a perspective view of a shape morphing elastic
tailored structure.
[0017] FIG. 7C is a perspective view of a shape morphing
multistable composite structure.
[0018] FIG. 8A graphically illustrates the force-displacement and
energy-displacement curves for a bistable mechanism.
[0019] FIG. 8B graphically illustrates the force-displacement and
energy-displacement curves for a metastable mechanism.
[0020] FIG. 9A compares the microstructure of nacre to brick and
mortar microstructure.
[0021] FIG. 9B schematically illustrates a comparison of brick and
mortar compression/tension mechanisms to nacre.
[0022] FIG. 10 graphically illustrates bistable/metastable cell
configuration that mimics the nacre compression/tension.
[0023] FIG. 11A schematically illustrates phase transforming unit
cells organized as a 2-dimensional array.
[0024] FIG. 11B schematically illustrates phase transforming unit
cells organized as a 3-dimensional array.
[0025] FIG. 12A schematically illustrates a quasi-hexagonal array
of phase transforming bistable cells having curved-beam walls.
[0026] FIG. 12B is an enlarged view of a curved-beam wall of FIG.
12A.
[0027] FIG. 12C is a schematic view of 2-dimensional arrays of
curved-beam triangular and rectangular cells, respectively.
[0028] FIG. 13 schematically illustrates a 2-dimensional array made
from linear cahins of phase-transforming cells.
[0029] FIG. 14 is graphically illustrates energy dissipation due to
hysteresis from loading/unloading phase-transforming cells.
[0030] FIG. 15A is a top plan view of a base-cell according to FIG.
1A.
[0031] FIG. 15B is a top plan view of a multicellular
phase-transforming array.
[0032] FIG. 16A graphically illustrates the force-displacement
relationship for a chain of 2 base cells of FIG. 15A.
[0033] FIG. 16B graphically illustrates the force-displacement
relationship for a chain of 6 base cells of FIG. 15A.
[0034] FIG. 16C graphically illustrates the force-displacement
relationship for a chain of 14 base cells of FIG. 15A.
[0035] FIG. 16D graphically plots .sigma./E vs. .epsilon. for the
array of FIG. 15B.
[0036] FIG. 16E illustrates the array of FIG. 15B under increasing
compression.
[0037] FIG. 17A schematically illustrates the forces on a curved
wall.
[0038] FIG. 17B schematically illustrates the force-displacement
relationship for a curved wall undergoing phase change.
[0039] FIG. 17C graphically compares a single walled bistable
mechanism to a reinforced double walled bistable mechanism.
[0040] FIG. 17D schematically illustrates triangular and square
multistable unit cells.
[0041] FIG. 17E schematically illustrates arrays of triangular and
square multistable unit cells.
[0042] FIGS. 17F-17K schematically illustrate the effects of
increasing temperature on thermally actuated bistable and
metastable phase transforming cells.
[0043] FIG. 18A is a plan view of a phase transforming cellular
material made up of curved-beam bistable unit cells.
[0044] FIG. 18B graphically illustrates the force/displacement
relationship of FIG. 18A during loading and unloading.
[0045] FIG. 19A-K graphically and schematically illustrate the
properties of thermal phase transforming cellular materials.
[0046] FIG. 20A graphically illustrates the force-displacement
relationship of a thermal phase transforming unit cell over a range
of temperatures.
[0047] FIG. 20B schematically illustrates the force-displacement
relationship of FIG. 20A over a range of temperatures.
[0048] FIG. 20C graphically illustrates the force/time/temperature
interrelationships for a thermal phase transforming unit cell.
[0049] FIG. 20D graphs the temperature-force relationship for
Fv.
[0050] FIG. 21A schematically illustrates the auxetic hexachiral
structure.
[0051] FIG. 21B schematically illustrates a unit cell of a
hexachiral phase transforming cellular material (h-PXCM).
[0052] FIG. 21C is a perspective view of a curved ligament of FIG.
21B.
[0053] FIG. 21D is a perspective view of the ligament of FIG. 21C
under bending.
[0054] FIG. 21E is a perspective view of the ligament of FIG. 21D
under reverse bending.
[0055] FIG. 21F graphically illustrates the relationship between
moment and applied angle for the ligaments of FIGS. 21D-E.
[0056] FIG. 22 is a view of a displacement measuring device for
chiral PXCMs.
[0057] FIG. 23A illustrates a cylindrical shell ligament h-PXCM
array prior to displacement testing.
[0058] FIG. 23B illustrates a flat ligament h-PXCM array prior to
displacement testing.
[0059] FIG. 23C illustrates a cylindrical shell ligament h-PXCM
array after displacement testing.
[0060] FIG. 23D illustrates a flat ligament h-PXCM array after
displacement testing.
[0061] FIG. 23E graphically illustrates the load/displacement
relationship for FIGS. 23A, 23C under loading and unloading.
[0062] FIG. 23F graphically illustrates the load/displacement
relationship for FIGS. 23B, 23D under loading and unloading.
[0063] FIG. 24 graphically illustrates the load-displacement
relationship for FIG. 21B for loading and unloading with and
without applied frequency mode imperfections.
[0064] FIG. 25A graphically illustrates the .pi..sub.1/.pi..sub.2
h-PXCM design space.
[0065] FIG. 25B graphically illustrates the energy dissipated per
unit volume for h-PXCM.
[0066] FIG. 25C graphically illustrates the relationship between
h-PXCM energy dissipated and .pi..sub.2.
[0067] FIG. 25D is a contour plot of average plateau stress of the
chiral PXCM as a function of .pi..sub.1.
[0068] FIGS. 26A-D plot analytical peak load, simulated peak load,
analytical plateau load, and simulated plateau load, respectively,
as functions of .pi..sub.1.
[0069] FIG. 27 graphically and schematically illustrates the space
defined as applied load .sigma. vs. work for several different PXCM
configurations.
[0070] FIG. 28A schematically illustrates two adjacently-disposed
curved PXCM chains.
[0071] FIG. 28B schematically illustrates a tire made of
adjacently-disposed curved PXCM chains.
[0072] FIG. 29 is a perspective view of an earthquake-resistant
plate incorporating PXCMs.
DETAILED DESCRIPTION
[0073] Before the present methods, implementations, and systems are
disclosed and described, it is to be understood that this invention
is not limited to specific synthetic methods, specific components,
implementation, or to particular compositions, and as such may, of
course, vary. It is also to be understood that the terminology used
herein is for the purpose of describing particular implementations
only and is not intended to be limiting.
Overview
[0074] The present novel technology relates to a novel cellular
material exhibiting discrete phase transformations. Phase
transformations are iniated by introducing changes to the geometry
of the unit cells that define these materials while keeping
topology constant. Phase transformations may be introduced into the
novel cellular materials via bistable/metastable compliant
mechanisms to form the microstructure of cellular materials.
[0075] The novel cellular material includes bistable or metastable
mechanisms as a unit cell for its microstructure. A bistable
mechanism has two stable configurations when unloaded. Once this
mechanism is in one stable configuration, it remains there unless
it is provided with enough energy to move to the other stable
configuration. A metastable mechanism corresponds to a special case
of stability in which a small disturbance can lead to another
stable state that has a lower potential energy. The phase
transformation capability of this new type of cellular material
will be attained mainly by proper choice of base material, cell
topology and geometrical design of the unit cell (see FIG. 1A). The
unit cell of the microstructure comprises a bistable mechanism in
which the two stable configurations correspond to stable
configurations of the phase transforming material. FIG. 1B
illustrates a combination of these unit cells to form a 1D periodic
cellular structure. Phase transformation occurs when there is a
progressive change of configurations from cell to cell leading to a
saw-tooth like force-displacement behavior. The same saw-tooth
pattern has been observed experimentally in the stress induced
phase transformation of a NiMnGa (see FIG. 2). Phase transforming
materials based in cellular solids allow for an increased
application of cellular materials in areas like wave guiding,
energy harvesting, energy dissipation and material actuation,
enabling new applications that were not possible before.
Wave Guiding Metamaterials
[0076] In conventional structural elements, such as plates, the
structural, noise, and vibration responses are strongly coupled
through shared design parameters (such as thickness). The
microstructure in a cellular solid can partially decouple these
responses, enabling the creation of structural elements with
inherently better noise and vibration mitigation properties than
conventional structural elements. The periodicity of the cellular
materials structure leads to two interesting phenomena concerning
the propagation of mechanical vibrations in these materials. First,
the frequency response of cellular materials show alternating
regions of marked attenuation (stop bands) and propagation (pass
bands). The former are characterized by vibration modes that are
localized to a unit cell, while the latter correspond to modes that
span multiple unit cells (see FIG. 3A). The transmission loss in
the stop bands is comparable to that achieved by conventional
damping treatment (e.g. constrained layer damping coatings) in
higher frequency ranges. Second, wave propagation in cellular
materials exhibit strong directionality. Waves can propagate more
easily along some favored directions than others (see FIG. 3B).
Moreover, the set of favored directions, as well as the amplitude
modulation within this set, is strongly dependent on the frequency
of the vibrations relative to the natural frequencies of the
component made from the cellular material. Hence, the wave
propagation behavior of the distinct stable configurations of a
phase transforming cellular material is significantly
different.
Energy Harvesting
[0077] Energy harvesting consists of capturing energy from external
sources (such as solar, wind, mechanical, and the like) and storing
said energy for later use. Recently, interested in this field has
been directed to kinematic energy generators which convert energy
in form of mechanical movement (vibrations, displacements, forces)
into electrical energy using electromagnetic, piezoelectric or
electrostatic mechanisms. Among the different alternatives for
kinematic energy harvesting, piezoelectrics have attracted
considerable attention for its capacity of provide continuous and
stable power supply. A cantilever structure with piezoelectric
material attached to the top and bottom surfaces has been a
traditional geometry for harvesting energy from vibrations,
however, such devices have a narrow bandwidth, hence practical
applications of such devices is difficult as the vibration
frequency often varies with time which results in a power
reduction. New devices based on snap-through buckling allow
frequency tune up enabling wide-bandwidth operations at an ambient
vibration frequency, resulting in highly efficient energy
harvesting (see FIG. 4). These new devices open the door for the
design of new multistable materials for energy harvesting
applications. Phase transforming cellular materials made of
piezo-responsive materials (bulk or coatings) enable massive
parallelization of energy harvesting and offer the potential to
integrate energy harvesting into the structure.
Energy Absorption
[0078] Materials with good energy absorption characteristics are
used for packaging fragile objects, personal protection equipment
such as helmets and blast protection panels for military vehicles,
and the like. These materials typically exhibit a long plateau in
the stress-strain response after the limit point is passed. With
the judicious selection of cell topology, cell geometry, wall
material, and relative density, cellular materials can be designed
to provide outstanding properties for energy absorption (see FIG.
5A-5B). Recent advances in the development of materials for energy
absorption have been based on the understanding of absorbing
mechanisms that are present in nature. A wide range of
bio-mechanical phenomena is attributed to bistability and
multistability behavior, such as the unfolding and folding of titin
protein and compression of sarcomeres in limb muscles. Systems that
show bistable and multistable behaviors have been shown to be
excellent for energy dissipation. Phase transforming cellular
materials have the potential to integrate energy absorption into
structural elements thereby reducing the need for add-on noise and
vibration mitigation treatments. One potential application could be
the use of phase transforming cellular materials in football
helmets to reduce impact-induced brain injuries. For such case,
both energy absorption and wave guiding properties may be
exploited.
[0079] From the energy absorption point of view, phase transforming
cellular materials enable new designs of passive energy dissipation
systems for seismic applications. Passive energy dissipation
devices are used in high rise buildings to limit damaging
deformations in structural components. These devices are grouped in
two broad categories: rate-dependent and rate-independent devices
(see FIG. 6A-6B). Rate-dependent devices consist mainly in dampers
whose force output is dependent of the rate of change of
displacements across the damper. Examples of these systems include
viscoelastic fluid dampers and viscoelastic solid dampers. The
disadvantages of rate-dependent devices include limited deformation
capacity, change of the properties of the viscoelastic component
due to frequency and temperature, and the maintenance cost
associated to wear of seals. On the other hand, rate independent
systems consist of dampers whose force output is not dependent on
the rate of change of displacement across the damper but rather
upon the magnitude of the displacement and possibly the sign of the
velocity. The behavior of these devices is described using
hysteretic models and examples include metallic and friction
dampers. Metallic dampers exhibit hysteretic behavior associated
with the yield of mild steel while friction dampers exhibit
essentially bilinear hysteretic behavior with very initial high
stiffness. The main disadvantage of rate-independent devices for
passive energy dissipations is that these systems suffer damage
after an earthquake and require to be replaced, also in the case of
friction dampers, the sliding interface may change with time. The
main characteristics that make of phase transforming cellular
materials good candidates for the design of passive energy
dissipation systems are: dissipation of energy due to hysteretic
behavior with recoverable deformations and large displacements,
long term reliability and stability to changes in environmental
conditions, such as temperature changes.
Shape Morphing Structures
[0080] A morphing structure refers to a system or assembly with the
ability to produce large deformations while bearing load.
Deployable structures such as those used in stadium roofs,
satellites, stents and vehicle roofs are common examples of
morphing structures. Approaches to creating morphing structures
include tensegrity structures (see FIG. 7A), elastically tailored
structures, active structures controlled by distributed
sensor-actuator systems (see FIG. 7B) and multistable composites
(see FIG. 7C). Elastically tailored structures are mainly based on
the concepts employed in compliant mechanisms in which transmission
of force and motion is obtained by deformation instead of the
connection of rigid bodies by joints. Morphing of a sandwich panel
with a Kagome truss core has been demonstrated by replacing some
truss members actuators. Active structures allow for large changes
in area, but they do not provide good load bearing capabilities.
Phase transforming cellular materials can be designed to exhibit a
large change in volume accompanying a change between stable
configurations while retaining sufficient mechanical properties for
structural applications.
Switchable Hydrophobic/Hydrophilic Surfaces
[0081] A switchable surface combines the attributes of both
superhydrophilic and superhydrophobic surfaces and can be used in a
variety of applications including microfluidic pumps, drug delivery
systems, windshield coatings, and protein concentrators. One method
to generate reversible wettability upon switching between
superhydrophobicity and superhydrophilicity is achieved by
biaxially extending and unloading an elastic polyamide film with
triangular net-like structure composed of fibers of about twenty
micrometers in diameter. The average side of the triangle of the
net-like structure is around two-hundred micrometers before biaxial
extending (superhydrophobic with a contact angle of 151.2 degrees)
and four hundred and fifty micrometers after extension
(superhydrophilic with a contact angle of 0.+-.1.2 degrees). The
mechanical actuation drives an increase in the liquid/solid
interface resulting in the modification of the apparent contact
angle rather than directly modifying the surface wetting
properties, which depend on the chemical composition of the
surface. A thin layer of a phase transforming cellular material can
be used in a similar manner wherein a change in phase leads to a
change in the apparent contact angle at the surface. If a bi-stable
unit cell is used in the phase changing material, an external
energy source such as an applied voltage can be used only to
transition the surface between its phases while no energy is needed
to maintain the current phase.
Choice of Unit Cell for Phase Transforming Cellular Materials
[0082] Extension of the notation of phase transformation of solids
to cellular materials can be obtained by the formation of
interconnected chains with a proper choice of a unit cell that
replicates the saw-tooth like load-displacement behavior
characteristic of a phase transformation. In this sense, the unit
cell is typically suitable to present snap-trough instabilities
when loaded, and when these instabilities at the cell level are
triggered it is considered that a phase transformation of the
cellular material has occurred. Snap-trough is typically associated
with a buckling instability in which at a certain point under
loading the force starts to decrease while the displacement is
increasing. In some cases, instabilities in the unit cells occur
during the elastic regime of the base material and can be
recovered, therefore unit cells based on bistable/metastable
compliant mechanisms are good candidates. Compliant mechanisms are
a type of mechanisms that employ elastic body transformations
instead of traditional joints to transfer force or motion.
Compliant bistable and metastable mechanisms are a particular class
of compliant mechanisms in which the force-displacement presents
three phases: in phase I the load increases with the displacements
until a critical load (F.sub.max) in which snap-trough instability
initiates, at this point phase II starts and the load keeps
decreasing until a minimum load (F.sub.min) at which phase III
begins and the load starts to rise again with displacement (see
FIG. 8A-8B). In the case of a bistable mechanisms there are two
stable configurations when unloaded. Once this mechanism is in one
stable configuration, it remains there unless it is provided with
enough energy to move to the other stable configuration (see FIG.
8A). Whereas a metastable mechanism corresponds to a mechanism that
is in a special case of stability in which a small disturbance can
lead to another stable state that exist nearby and has a lower
potential energy (see FIG. 8B).
[0083] In order to use bistable/metastable compliant mechanisms as
a base for the unit cell of a phase transforming cellular material,
it is helpful to form a microstructure that allow the boundary
conditions on each mechanism to be maintained so as to allow the
change from one stable configuration to the other upon loading of
the cellular material. The microstructure of nacre provides a
bio-inspired solution to maintain these conditions. This
microstructure is formed by a wavy "brick and mortar system" (see
FIG. 9A). The wavy bricks, in conjunction with the mineral bridges
and the organic glue activate a very unique competition between a
compression and a tension mechanism. When the tension mechanism
wants to expand, the compression mechanism shrinks preventing the
expansion (see FIG. 9B). These competition mechanisms can be
approximated by the introduction of stiff horizontal walls to a
regular hexagonal cellular microstructure (see FIG. 10A). Under
compressive loading, the regular hexagonal material will try to
expand but the stiff walls will prevent that expansion. These
competition mechanisms will allow the inclined walls that form the
regular hexagonal microstructure to act as a bistable/metastable
mechanism, therefore under continuation of the compressive loading,
the material will typically undergo the transformation to the
second phase that correspond to an inverted hexagonal configuration
(see FIG. 10B).
[0084] The idea of competition mechanisms and the curved beam
mechanisms as a base for phase materials can be extrapolated to
produce bi-directional instabilities in 2D plane honeycombs and 3D
foams. FIG. 11A shows the 2D structure where the triggering of
stable configurations has two preferential directions. The same
concept applies for a 3D structure shown in FIG. 11B. In order to
reduce the stress concentration at the joints of the hexagonal
phase transforming material shown in FIG. 10, the inclined walls of
the mechanism are replaced by a curved beam mechanism (see FIG.
12A). The typical implementation of this mechanism consists of a
cosine shaped monolithic beam described by the thickness t, depth
b, wavelength .lamda., amplitude A. the cosine shape of the curved
beam mechanisms is given by equation 1 (see FIG. 12B).
A _ .function. ( x ) = A .function. [ 1 - cos .function. ( 2
.times. .pi. .times. x .lamda. ) ] ( 1 ) ##EQU00001##
In order to obtain snap-trough, the geometry constant Q=A/t is
typically greater than or equal to 6 and the behavior a single
curved mechanism can be approximated by:
F max = 7 .times. 4 .times. 0 .times. EIA .lamda. 3 .times. d max =
0 . 1 .times. 6 .times. A ( 2 ) F min = 3 .times. 7 .times. 0
.times. EIA .lamda. 3 .times. d min = 1.92 .times. A ( 3 )
##EQU00002##
where E corresponds to the Young's modulus of the mechanism
material and I is the moment of inertia of the curved beam. To
ensure an elastic and reversible change between phases, the maximum
strain of the curved beam during the deflection should be kept
below the yield point of the material. The maximum strain during
deflection can be estimated by using equation 3.
max = 2 .times. .pi. 2 .times. t .times. A .lamda. 2 ( 4 )
##EQU00003##
Mechanical Behavior of Phase Transforming Cellular Materials
[0085] A construction of a phase transforming material may be
obtained by assembling an array of N.sub.c chains composed by Nunit
cell mechanisms in series (see FIG. 13). Using equations 1-3, the
displacement as a function of the force of the base mechanism on
each phase can be approximated using a linear system defined
between F.sub.min and F.sub.max as follows:
.times. Phase .times. .times. I .times. : .times. .times. x I
.function. ( F ) = F k 1 .times. .times. for .times. .times. x I
< d max .times. .times. Phase .times. .times. II .times. :
.times. .times. x II .function. ( F ) = F + k 2 .times. d max - F
max k 2 .times. .times. for .times. .times. d max < x II < d
min .times. .times. Phase .times. .times. III .times. : .times.
.times. x III .function. ( F ) = F + k 3 .times. d min - F min k 3
.times. .times. for .times. .times. x III > d min ( 5 )
##EQU00004##
Under the assumption that the system is overdamped (i.e. all the
excess of energy from the change of phase of a mechanism is
dissipated into heat) and defining P.sub.I, P.sub.II, and P.sub.III
as the number of mechanisms in Phase I, Phase II and Phase III
respectively, the force-displacement behavior for a chain can be
found using:
X(F)=P.sub.Ix.sub.I(F)+P.sub.IIx.sub.II(F)+P.sub.IIIx.sub.III(F)
with F.sub.min<F<F.sub.max and P.sub.I+P.sub.II+P.sub.III=N
(6)
Then, the total force-displacement behavior on the materials is
given by
X.sub.T(F)=N.sub.cX(F) (7)
A special case in the force-displacement behavior of a phase
transforming cellular material occurs when
(P.sub.I/k.sub.1+P.sub.III/k.sub.3)<|k.sub.2|, in which case
there is a hysteresis between the loading and unloading path of the
material due to snap-back instabilities. This hysteresis produces
energy dissipation that can lead to novel energy absorption
mechanisms (see FIG. 14). In general, structural cellular materials
absorb energy via a plastic deformation mechanism. However, a phase
transforming cellular material may absorb energy via recoverable
elastic deformation, allowing the material to bounce back to its
original configuration after the load is released.
[0086] Loss coefficient (.eta.) is a dimensionless parameter used
to characterize the energy that a material dissipates by intrinsic
damper and hysteresis. .eta. is defined in equation 8 and
corresponds to the energy loss per radian divided by the maximum
elastic strain energy (U).
.eta. = .DELTA. .times. U 2 .times. .pi. .times. U ( 8 )
##EQU00005##
Typically, for material selection in engineering applications the
loss coefficient is related to the damping ratio (.zeta.) by
using:
.zeta. = .eta. 2 ( 9 ) ##EQU00006##
Materials and Methods
Experimental
[0087] Prototypes of the base unit cell mechanism and phase
transforming materials were manufactured a multi-material 3D
printer using a rigid material as a base (see FIG. 15). The tensile
properties of the base material are listed in table 1.
TABLE-US-00001 TABLE 1 Properties of base material Property Unit
Metric Modulus of elasticity MPa 1053.83 .+-. 58.036 Yield strength
MPa 19.306 .+-. 1.284 Strain at yield [ ] 0.0485 .+-. 0.00213
Ultimate tensile strength MPa 18.0976 .+-. 0.3277 Strain at break [
] 0.3209 .+-. 0.036167
Computational
[0088] Computational simulations of the base unit cell and the
phase transforming cellular material were made using explicit FEM
under displacement control boundary conditions. Plane strain
elements with four nodes and four integration points were used
(CPE4) for the meshes of the models.
RESULTS
[0089] FIGS. 16A-16C present a comparison between predictions using
the analytical model given in equation 6 and simulations for chains
formed by 2, 6 and 14 double curved beam mechanisms in series. The
geometrical parameters considered in the base mechanisms are:
t=0.866 mm, b=10 mm, .lamda.=58.47 mm and A=10 mm. In general, a
good agreement is founded between the numerical and the analytical
model, but as the number of unit cells in the chain increases,
instabilities introduce nonlinearities that are not captured for
the analytical model. In the case of FIG. 16C, enough mechanisms
are stacked to produce snap-back behavior.
[0090] FIG. 16D-16E illustrate a comparison between experiments and
simulations for a multistable programmable material. In general,
good agreement was found between simulations and the experiment.
The geometry of the unit cell was designed such that the experiment
can be run in compression and consists on a single curved beam
mechanism. The single beam mechanism produces metastable behavior;
therefore, the cells recovered its original size when the load was
released. Another observation made experimentally (and confirmed by
simulations) is that the cell transformation from one state to the
other takes place one row at the time. The geometry of the unit
cell considered for this material is given by: t=0.72 mm, b=10 mm,
.lamda.=60 mm and A=5.04 mm, T=1.08 mm. Using the same parameters
for the base mechanism, a bigger material consisting of 12.times.4
cells was printed and tested for loading and unloading (see FIG.
18). This material exhibited hysteresis as predicted by the use of
equation 7. Also, a damping ratio of 7% was calculated for the
experiment using equation 9.
Thermal-Induced Recovery Phase Transforming Cellular Materials
[0091] Phase transforming cellular materials (PXCMs) are a subset
of the architectured materials discussed above whose unit cells
have multiple stable configurations and can absorb energy by
allowing non-equilibrium release of stored energy through
controlled elastic limit point transitions as the cells transform
between different stable configurations. Prior art materials with
elastic limit point transitions have focused on material behavior
under mechanical loading. The forward transformation in these
materials always happens under an applied mechanical load, while
the reverse transformation can be driven either by elastic energy
stored in the material during the forward transformation
(metastable PXCMs) or by an external force acting in the direction
opposite to that of the force applied during the forward
transformation. The novel PXCM differ insofar as the forward
transformation still happens under an external applied force but
the reverse transformation is driven by a thermal stimulus (see
FIGS. 17A-17E AND 19A-19K). A family of bi-material PXCMs
exhibiting this behavior is discussed below. FE simulations in
addition to examples of 3D printed samples illustrate the
underlying mechanics.
[0092] The novel PXCMs may be envisioned as programmable
metamaterials that can mimic the shape memory effects of Shape
Memory Alloys (SMAs), and which may be tuned to have geometrical
phase transformation through physical stimulus. Having tunable
phase transformation enables these materials applications of
actuation, energy harvest, and energy dissipation. Temperature
variation on thermal PXCMs changes their heterogeneity, triggering
phase transformation in addition giving rise to shape memory
effect. Therefore, thermal PXCMs have potential on creating auction
and energy harvest devices. Many studies have been investing
manmade metamaterials to achieve this shape memory effect.
Thermal PXCMs Design
[0093] To mimic the shape memory effect of SMAs, three thermal
PXCMs designs have been identified based on 1D PXCMs. Each unit
cell contains a bent beam, stiffer walls at center, ends and center
(see FIG. 17A). The shape of the bent beam is described by the
expression
Y = ( A 2 ) .function. [ 1 - cos ( 2 .times. .pi. .times. X /
.lamda. ] ( 10 ) ##EQU00007##
Q=A/t determines the bistability each unit cell. A force may be
applied to a unit cell to cause it to transform from Phase I into
Phase II. When Q>2.31, release this force, the bent beam remains
in the Phase II. When Q<2.31, the bent beam recovers back to
Phase I. Distinct from 1D PXCMs, bistability of thermal PXCMs is
not only depends on the geometry properties but also surrounding
temperature. At low temperature, thermal PXCMs are bistable and at
high temperature, they are metastable. Therefore, thermal stimulus
can trigger phase transformation.
[0094] All three designs are composed of two types of base material
with different thermomechanical properties. Most portion of a unit
cell is made of a material (M.sub.1) which is not sensitive to the
thermal stimulus. A small portion of the mechanism is made of the
material (M.sub.2) whose mechanical properties reduce dramatically
when the temperature increases. M.sub.1 and M.sub.2 have similar
mechanical properties at the low temperature. As the result,
although PXCMs comprise two types of materials, they are
approximately homogeneous at low temperature. Therefore, at low
temperature, Q is still the factor that governs bistability of
thermal PXCMs. When Q>2.31, thermal PXCMs can be bistable at low
temperature. While temperature increasing, the mechanical
properties of two materials departure rapidly. Once the temperature
exceeds a critical value, the mechanical properties M.sub.2 become
quite low compared with M.sub.1. This heterogeneity of the
materials causes the mechanism becomes metastable.
Analytical Model
[0095] The analytical model is helpful to understand and design the
Type III thermal PXCMs which can recover at desired temperature. It
is created based on the analytical model of 1D PXCMs (FIGS. 17A-17E
and 19A-19K). Type III thermal PXCMs comprise a numerous of
periodically arranged unit cells. Each unit cell is composed of a
bent beam and stiffer walls at bottom, top, and ends. M.sub.2,
which is assigned on the boundary stiffer walls, has the elastic
modulus varies rapidly with temperature. It provides different
levels of transitional constraints based on the temperature.
Therefore, the unit cell of type III thermal PXCMs is modeled as a
bent beam with one end clamped and the other end series connected
with a translational spring. The stiffness of this translational
spring varies from rigid to zero when temperature increase from low
to high. When the stiffness of the translational spring approaches
infinity, the model is equivalent to a bent beam with
clamped-clamped boundary condition, which is the analytical model
of 1D PXCMs. When the stiffness of the translational spring
approach to zero, the model is equivalent to a bent beam clamped at
one end and free to extend at the other end. The force-displacement
relation of type III thermal PXCMs are shown. The performance of
mechanism under mode 1 and mode 3 are given as F.sub.1 and
F.sub.3-d relations. N is dimensionless parameter after normalizing
the axial force. c is the coefficient that describes the stiffness
of translational spring provided to the unit cell. F.sub.1 depends
on both parameters Q and C. F.sub.3 is only linear depends on d.
The intersections between F.sub.1-d and F.sub.3-d are where the
mechanism switch from mode 1 to mode 3. When the translational
spring stiffness k.sub.s approaches to be rigid, c becomes
infinitely close to 1. When c decreases, the bistability of the
unit cell decrease.
Design Based on Analytical Model
[0096] Under the temperature when two materials have similar
mechanical properties, the parameter determines the bistability of
thermal PXCMs is Q, which is same for 1D PXCMs. Under other
temperatures, c is the parameter determines bistability of thermal
PXCMs. There is a critical value c at the point when a bistable
mechanism transfers into a metastable mechanism. This critical
value is defined as the parameter c.sub.critical. Every Q of a bent
beam has a c.sub.critical value which captures the transition from
bistable to metastable. FEA and analytical models are used to
obtain the c.sub.critical corresponding to different Q. They are
plotted together in FIG. 19E and show agreement.
[0097] For a bent beam with given Q, the temperature of any
geometry and thermomechanical combinations of bent beam and stiffer
walls that satisfies c=c.sub.critical is the recovery
temperature.
F 1 = j = 1 , 5 , 9 , 13 .times. .times. .infin. .times. 4 .times.
( N 2 - N 1 2 ) N j 2 .function. ( N 2 - N j 2 ) 2 .times. F 1 2 -
N 1 2 .times. F 1 + N 2 .function. ( N 2 - N 1 2 ) 2 1 .times. 2
.times. Q 2 .times. c - N 1 2 .times. N 2 .function. ( N 2 - 2
.times. N 1 2 ) 1 .times. 6 = 0 ( 11 ) .times. F 2 = 1 .SIGMA. j =
1 , 5 , 9 , 13 .times. .times. .infin. .times. 8 N j 2 .function. (
N 2 2 - N j 2 ) 2 .times. ( N 2 2 N 2 2 - N 1 2 - .DELTA. ) ( 12 )
.times. F 3 = 1 .SIGMA. j = 1 , 5 , 9 , 13 .times. .times. .infin.
.times. 8 N j 2 .function. ( N 3 2 - N j 2 ) 2 .times. ( N 3 2 N 3
2 - N 1 2 - .DELTA. ) ( 13 ) .times. N 2 = p .times. l 2 EI ( 14 )
.times. c = 1 k a / k s + 1 ( 15 ) ##EQU00008##
Experiments and FEA Validation
[0098] The analytical equations suggest that for given materials
M.sub.1and M.sub.2, type III thermal PXCMs can be designed to have
shape memory effect. The recovery temperature depends on the
c.sub.critical which is determined by the geometry details of bent
beam and stiffer walls. To evaluate this design concept, FEA
simulation and experiment are conducted on a prototype (FIG. 19C).
Grey 60, which has less sensitivity to the temperature variation,
is used as M.sub.1. Shore 95 is used as M.sub.2 since its elastic
modulus decreases rapidly when temperature increases (FIG. 19D).
FIG. 19E illustrates how thermal PXCMs with same wavelength and
different amplitude to recovery at different temperature by using
these two materials. More examples are demonstrated in
supplementary materials.
[0099] The prototype was fabricated by multi-material polymer
printer. Both FE model and specimen are shown in FIG. 19F. We
design the recovery temperature of this specimen is 18.degree. C.
The thickness of the stiffer walls are designed to ensure
c=c.sub.critical at this recovery temperature 18.degree. C. The
specimen is placed on an aluminum 8020 frame. Two L shape angles
are fixed on both side of the specimen to eliminate the move in X
direction. The testing procedure is as show as follow, (1) Compress
the prototype to transfer from Phase I to Phase II under 8.degree.
C. (2) Let the sample remains in Phase II under the same
temperature for 10 minutes to eliminate the shape recovery effects
from viscosity. (3) After 10 minutes increase temperature gradually
to 22.degree. C. FE simulation and experiments are shown from FIG.
19G-19K. As expected, the specimen can remain bistable for 10
minutes at 8.degree. C. Therefore, viscosity of base materials did
not play the role in recovery. As temperature increase, there are
no obvious recovery could be observed. When the temperature
increases to 18.degree. C. the bottom stiffer bar of the prototype
started to lift and eventually at 19.degree. C. the specimen is
fully recovered.
[0100] Thermal PXCMs exhibit shape memory effects from both FEA
simulation and experiment. Furthermore, we investigate whether
thermal PXCMs can be used as thermal-actuator devices like SMAs.
FIG. 20A shows the F-d relation of a thermal PXCM unit cell under
the temperature varies from 5 to 30.degree. C. It is observed that
the valley force F.sub.v increases from negative to positive when
temperature increases. At 15.degree. C., the unit cell becomes
metastable and F.sub.v=0 N. If a weight W is applied on the unit
cell, the transition would happen when the F.sub.v=W N. Therefore,
higher temperature is required to achieve phase transformation if a
unit cell under a dead load.
[0101] The weight a mechanism can lift is equivalent to the valley
force of a type III thermal PXCMs under the recovery temperature. A
test procedure is facilitated as below. [0102] 1. Transfer a
thermal PXCM unit cell from BP.sub.1 to BP.sub.2 under an initial
temperature where two materials have relatively close mechanical
properties. [0103] 2. After the mechanism transformed, released the
force. [0104] 3. Applying a weight to the unit cell. [0105] 4.
Increase the temperature until the mechanism recovery with the
weight block. [0106] 5. Applying the force on a mechanism under
this recovery temperature [0107] 6. Record the valley force F.sub.v
of the mechanism under this recovery temperature [0108] 7. Compare
the applied weight W and valley force F.sub.v. FIG. 20B
demonstrates how the stiffer walls response during this process. At
point 0 stiffer walls are located at an original position. When the
force applied to the mechanism, the stiffer walls are compressed
due to the axial force during the phase transformation. The
contraction of stiffer walls reaches the maximum value (point 2) at
the peak load F.sub.p and then started to reduce. From point 3 to
5, the axial force caused by phase transformation reduces therefore
the deformation of stiffer walls release gradually. When the force
is released, the stiffer walls are compressed more to hold the bent
beam in BP.sub.2. When a weight is applied, the stiffer walls
become less compressed. When the temperature increases the
deformation of the stiffer walls increase again because reduction
of the stiffness. Eventually, the mechanism will recovery back to
its original configuration since not enough constrain can be
provided from the stiffer walls. This process happened by changing
the stiffness of the constraints.
[0109] A number of simulations are created followed by this test
procedure are shown in FIG. 20D. The valley forces of the mechanism
under different temperature are plotted by the red dots and the
weights of a mechanism can lift among different temperature are
plotted by black dots. The two curves almost overlap each
other.
Chiral Honeycombs with Phase Transformations
[0110] A phase transformation in a cellular material corresponds to
the change in geometry of its unit cell from one stable
configuration to another stable (or metastable) configuration while
keeping its original topology. The capability of a cellular
material to undergo phase transformation is attained mainly by a
proper choice of an elastic base material and the topological and
geometrical design of the unit cell in order to allow the elastic
reversibility of the transformation. Cellular materials that
exhibit phase transformations show hysteresis and their response is
characterized by long, serrated loading and unloading plateaus,
making these materials attractive for energy absorption
applications. In this section, a new class of phase transforming
cellular material (PXCM) based on a hexachiral motif is introduced
and discussed. In this new PXCM, the ligaments of a regular
hexachiral honeycomb are replaced by segments of cylindrical
shells. These segments themselves exhibit a metastable snap-through
under compression. The energy dissipation behavior of PXCMs with
ligaments that only exhibit elastic buckling has been shown to
exhibit size dependence--the dissipation behavior is only seen in
samples that comprise a minimum number of unit cells. Unlike these
PXCMs, the PXCMs presented in this work use the inherent
snap-through behavior of their ligaments to exhibit energy
dissipation behavior even in samples as small as one unit cell. The
hexachiral PXCMs presented here also constitutes a material that
exhibits phase transformation in any loading direction in the plane
of the sample, thereby corresponding to the first real 2D PXCM. The
novel PXCMs are approached with a combined framework that includes
analytical, experimental and computational analysis. From these
analyses, the hexachiral PXCM was been observed to exhibit energy
dissipation and hysteresis without dependence on size effects or
plastic deformation. In addition, the hexachiral PXCM requires a
relatively low plateau stress in order to achieve relatively large
energy dissipation, giving this material a new location on the
Ashby Plot.
[0111] Architectured materials known as phase transforming cellular
materials are composed of periodic unit cells, each of which
consisting of a compliant snapping mechanism that can be metastable
or bistable. The load-displacement behavior of each constitutive
unit cell is characterized by three fundamental regimes bounded by
two limiting points, (d.sub.I, F.sub.I) and (d.sub.II, F.sub.II)
known as the critical displacements and loads. Regimes I and III of
a unit cell display a positive stiffness, since these two regimes
represent the material undergoing deformation whilst in its stable
configurations. These stable configurations correspond to a local
minima of the potential energy in the unit cell. Regime II displays
a negative stiffness which corresponds to the unit cell
transitioning between the limiting points mentioned above.
[0112] Each of the unit cells in a PXCM utilize the snap-through
instability of its constituent mechanism. A snap-through
instability is a phenomena which is only achieved by structures
that exhibit snap-back upon loading. Snap-back occurs when a
structure experiences a reversal in displacement along with a
reduction in equilibrium force, which is required by the material
to induce a phase transformation. The instabilities induced in
PXCMs such as these allow these materials to dissipate energy over
the course of a loading and unloading cycle. However, in order to
achieve snap-through and thus energy dissipation, the effective
stiffness of the portion of a PXCM in regimes I and III must
satisfy the following condition:
( n I k I + n III k III ) - 1 .ltoreq. k II ( 16 ) ##EQU00009##
where n.sub.1 is the number of unit cells in regime I, k.sub.I is
the stiffness of each unit cell in regime I, n.sub.III is the
number of unit cells in regime III, k.sub.III is the stiffness of
each unit cell in regime III, and k.sub.II is the stiffness of each
unit cell in regime II. Thus, there is a size effect imposed on
previous PXCM that effects whether or not they can dissipate
energy. Size effects in previous PXCM geometries can also effect
plasticity and fracture. In addition, the performance of previous
PXCM geometries depended greatly upon the choice of loading
direction. However, the dependence on size effects comes from the
choice of the PXCM topology as well as the snapping mechanism.
[0113] Herein, an isotropic phase transforming cellular material
known as the hexachiral PXCM is introduced. The hexachiral PXCM
(h-PXCM) utilizes a periodic, chiral topology which consists of a
network of unit cells arranged in a hexagonal pattern. Topologies,
such as that of the chiral honeycomb, which are isotropic and
auxetic along any loading direction. The unit cell consists of
cylindrical supports connected via six cylindrical shell ligaments,
which are equipped with a transverse curvature orthogonal to the
axis along their length. The cylindrical shell ligaments are the
snapping mechanisms utilized by the h-PXCM, which can exhibit
snap-through instabilities elastically without any dependence upon
size effects. The ligaments were rigidly constrained to the
supports at their points of contact. The supports were left free to
move and rotate only in response to load applied to the ligaments.
It is interesting to note that each support was connected rigidly
to six ligaments, three of which were oriented concave up, while
the other three were oriented concave down. This was done to ensure
that the h-PXCM would consist of identical, repeating unit
cells.
The Hexachiral PXCM
Geometry and Dimensionless Groups
[0114] The general geometry of the h-PXCM unit cell (FIG. 21B) is
governed by several physical parameters including: the radius of
the supports, r, and the length, L, the angle of curvature,
.theta., the thickness, t, and the radius of curvature, R, of the
ligaments. However, by utilizing the Buckingham Pi Theorem, the
physical description of the h-PXCM could be generalized to the
following two dimensionless parameters: .pi..sub.1=L/.rho.,
.pi..sub.2=r/L, which are termed the slenderness ratio and the
ratio of circular support respectively. The slenderness ratio is
the relationship between the length of the ligaments and their
radius of gyration, .rho.. The radius of gyration is given simply
as: .rho.= {square root over (I/A)}, where I is the moment of
inertia of the cylindrical shell ligament and A is the
cross-sectional area of the ligament.
[0115] The h-PXCM exploits the snap-through instability of
cylindrical shell ligaments under uniform bending, which can be
observed graphically in FIG. 21F with the solid black arrows.
Cylindrical shell ligaments can snap-through without any dependence
on size effects. However, a cylindrical shell ligament with its
cross-section oriented concave down and bending shown in FIG. 1d
will not exhibit snap-through when subject to uniform bending.
Conversely, a similar cylindrical shell ligament with an identical
orientation but opposite bending presented in FIG. 21E will exhibit
snap-through. In response to a global compressive loading, the
cylindrical shell ligaments of the h-PXCM will experience snap-back
and consequentially snap-through without any dependence on size
effects or loading direction since the material is isotropic.
Structures such as the h-PXCM which consist of snapping mechanisms
that phase transform will exhibit irreversible energy dissipation
over the course of a loading and unloading cycle as well as
non-smooth changes in over-all volume. It is interesting to note
that for the purpose of these analyses, the cylindrical shell
ligaments were assumed to be under uniform bending.
Examples
[0116] To investigate the energy absorbing capabilities of the
h-PXCM, two samples were subjected to loading and unloading cycles
under displacement control in an MTS (Materials Testing System)
machine using a 10 kN load cell. One sample was fabricated using a
tape measure for the cylindrical shell ligaments. Flat steel sheets
were used to fabricate flat ligaments for the second sample. The
supports for both models were cut from hollow cylindrical aluminum
beams. The flat and cylindrical shell ligaments were all made of
steel. The flat ligaments were approximately 80 mm by 19 mm and had
a thickness of 0.23 mm. The cylindrical shells had a length of 80
mm, a radius of curvature of 11.25 mm, an angle of curvature of 106
degrees, and a thickness of 0.17 mm. The ligaments were screwed
into steel cylinders at the appropriate angles to create a
hexachiral structure. The cylindrical supports for both models had
an outer diameter of 25.4 mm and an inner diameter of 20.32 mm.
[0117] These experimental samples can be viewed in FIG. 23A-23B.
Each h-PXCM sample was placed on top of an 8020 beam, which was
held in place by a grip, which was itself rigidly attached to the
bottom of the MTS machine (not shown). An 8020 beam was placed on
top of the sample, which was connected to the 10 kN load cell (not
shown) via a grip. Two 8020 beams were placed on the front and back
faces of the PXCM and were secured in place via connections to the
bottom 8020 beam for the purpose of avoiding out of place buckling
in the sample. These beams also made it possible to experiment with
a roller connection between the cylindrical supports of the PXCM
samples and the top and bottom 8020 beams. Since the hexachiral
PXCM was metastable and deformed elastically, no connections were
made between the top and bottom 8020 beams and the sample. A strain
rate of 1 mm/min was applied to the top of each h-PXCM sample
during the tests.
[0118] The cylindrical shells composing the h-PXCM in FIG. 23A
exhibited snap through instabilities and as a result hysteresis,
which can be observed in FIG. 23C. These instabilities are the
result of each cylindrical shell ligament bending in response to
the global loading applied by the MTS machine, which allows the
material to dissipate energy. In the case of the sample shown below
in FIG. 23A, the energy dissipated (extracted from the load
displacement curve shown in FIG. 23E) was found to be approximately
815 mJ, which was approximately 34.5% of the energy that was put
into the system upon loading. The flat ligament h-PXCM shown in
FIG. 23B did not exhibit snap through behavior. However, the sample
was observed to dissipate approximately 2086 mJ of energy, which
was approximately 18.4% of the energy that was put into the system
upon loading.
[0119] Under ideal conditions, the flat ligament h-PXCM structure
would not dissipate energy and any hysteresis exhibited by the load
displacement curve shown in FIG. 23F is not due to instabilities
within the structure, but rather due to friction between the
experimental model and the MTS machine. The same errors also occur
within the h-PXCM with curved ligaments, however, under ideal
conditions this model would still exhibit snap-through
instabilities resulting from the cylindrical shell ligaments phase
transforming. These phase transformations are the result of
irreversible energy dissipation by the material accompanied with a
non-smooth change in volume upon loading and unloading. These
experiments demonstrated that an h-PXCM design composed of
cylindrical shell ligaments equipped with a non-zero curvature will
have a larger percentage of energy dissipated than an equivalent
h-PXCM design equipped with flat (zero-curvature) ligaments. For
the purpose of further analysis, a parametric analysis was
performed on the h-PXCM unit cells energy dissipation
capabilities.
[0120] In addition, several other designs were tested such as the
tetra-antichiral PXCM and the tetra chiral PXCM. Fundamentally,
hexachiral geometry was chosen since this structure exhibited no
plastic deformation after several loading and unloading cycles,
while the other two geometries mentioned above did exhibit plastic
deformation over the course of multiple loading and unloading
cycles.
Finite Element Models and Parametric Analyses
[0121] To support the experimental results displayed in the
previous section, a parametric analysis of the energy dissipating
capability of the h-PXCM unit cell was performed using a series of
twenty-one finite element (FE) models, which were designed using
the two dimensionless parameters .pi..sub.1 and .pi..sub.2. For
each of the models, the curvature was held constant, in addition to
the length of the ligaments. The mass, radii of the cylindrical
supports, and the angles of curvature of the cylindrical shell
ligaments were varied by model. Each unit cell FE model was
constructed using S4R shell elements and steel as a base material
for the cylindrical shell ligaments. The cylindrical supports were
modeled with a higher elastic modulus than steel to ensure that the
supports were more rigid than the ligaments. A schematic of an
example h-PXCM unit cell FE model can be seen in FIG. 1b. Periodic
boundary conditions were applied appropriately to the unit cell FE
models through the use of a dummy node that was not part of the
model geometry. Whatever boundary conditions that were applied to
the dummy node, were also applied in a similar way to the FE
models. Additionally, it is interesting to note that the global
angle considered for the finite element simulations was different
than that of the repeated unit cells in the experimental samples
displayed in FIG. 2a-b. The orientation of the h-PXCM unit cell FE
models was chosen to simplify the unit cell modeling process.
Additionally, the orientation of the h-PXCM experimental samples
was chosen since this was the easiest orientation for testing in
the MTS machine.
[0122] Violent snapping occurred when the cylindrical shell
ligaments in the unit cell FE models were loaded and exhibited
snap-back and snap through. These violent events would often cause
the simulations to crash due to the requirement of a small time
increment, thus imperfections were applied to each model in the
form of a summation of the first fifteen modes of vibration of the
unit cell FE model. To do this, a frequency analysis was performed
on each model to extract the modes of vibration which were then
weighted, summed together, and then applied to the FE models as
imperfections. The purpose of the application of these
imperfections was to simulate a slightly crinkled cylindrical shell
that would not buckle quite as violently upon loading as a perfect
cylindrical shell ligament. To ensure that the imperfections were
not drastically changing the results, an unit cell FE model that
finished through a loading and the unloading simulation without the
imperfections, was compared to its counterpart with the appropriate
frequency modes applied as imperfections. The differences in the
resulting energy dissipated and the average plateau stress was
found to be negligible (FIG. 24).
[0123] From these loading and unloading simulations, the energy
dissipated per unit volume by each of the unit cell FE models was
as well as the average plateau stress was extracted. A strain rate
of 10 mm/s was applied to the dummy node, which loaded the unit
cell FE model while accounting for the periodic boundary
conditions. The energy dissipated per unit volume and the average
plateau load were extracted as points for each of the models with
different .pi..sub.1 and .pi..sub.2 and plotted on an Ashby Plot to
compare the h-PXCM to the energy dissipating capability of other
materials under displacement control. The energy dissipated by the
unit cell FE modeled without imperfections was approximately 371
mJ. The same FE unit cell model applied with the frequency mode
imperfections dissipated approximately 373 mJ.
Analytical Equations
[0124] Consider an h-PXCM composed of infinitely many unit cell's.
The forces and moments applied to each of the unit cell's
cylindrical shell ligaments can be approximated via a free body
diagram. Note that in the formulation of the analytical equations
for the h-PXCM unit cell, two fundamental assumptions were made:
[0125] (1) Each cylindrical shell in the h-PXCM unit cell is
assumed to bend due to compressive loads applied along the axis to
the ligament length before the moments take effect. [0126] (2)
Bending occurs simultaneously in each of the cylindrical shell
ligaments
[0127] The load, P that is felt by the h-PXCM unit cell under a
compressive loading is given by the following general
expression:
P=F.sub.lig(-cos(.beta.+.theta.')+cos(60.degree.+.beta.+.theta.!)+sin(.b-
eta.-30.degree.+.theta.')) (17)
Where F.sub.lig is the axial load, which is assumed to be
equivalent in each ligament, which acts on each of the ligaments in
the unit cell. The angle .theta.' is the global offset angle
between the global coordinate system and the unit cell coordinate
system. In the case of the simulated h-PXCM models .theta.'=30
degrees. For the experimental h-PXCM samples .theta.'=0
degrees.
[0128] Using these assumptions and the general expression for the
load, P felt by the unit cell given in Eqn. 17, it is possible to
derive an expression for the peak load of the RVE. This can be done
by utilizing the Euler buckling formula, which represents load
force required to bend the cylindrical shell ligaments
individually. The Euler Buckling Formula is given by the
following:
P crit = .pi. 2 .times. EI L 2 ( 18 ) ##EQU00010##
Where the moment of inertia, I applies to the cylindrical shell and
is given as the following.
I = .theta. .times. .times. tR 3 .function. ( 1 - 2 .times. ( sin
.function. ( .theta. ) .theta. ) 2 + ( sin .function. ( 2 .times.
.theta. ) 2 .times. .theta. ) ) ( 19 ) ##EQU00011##
The load required to induce bending in each of the cylindrical
shells ligaments in the unit cell, which is termed the critical
load, F.sub.crit was acquired by substituting Eqn. 18 in for
F.sub.lig in Eqn 17.
[0129] The resulting critical load is given as the following.
F.sub.crit=P.sub.crit(-cos(.beta.+.theta.')+cos(60.degree.-.beta.+.theta-
.')+sin(.beta.-30.degree.+.theta.')) (20)
In addition, an expression for the plateau load, which can be used
to determine the plateau stress of the h-PXCM, can be deduced. To
formulate this expression, consideration was given to the moments
that act on each of the ligaments due to the axial loads imposed on
them. For the purposes of the analyses conducted here, the moments
applied to each of the cylindrical shell ligaments, are all assumed
to be equivalent. Any rotations observed in the cylindrical
supports are a consequence of the moments applied to them by the
ligaments. These moments also contribute to the bending that occurs
in the ligaments. To fully understand the bending behavior of the
cylindrical shell ligaments, the relationship between the moment M
applied to the end edges of a cylindrical shell ligaments and the
applied angle .PHI..sub.a was analyzed. Note that .PHI..sub.a is
the angle between the x-axis and the axis running along the
ligament's length. For our analysis, we chose to use the same sign
convention introduced in for the moments and angles applied to a
cylindrical shell ligament. A negative moment and applied angle
induces no snap-through in the ligament whereas a positive moment
and applied angle will induce a snap-through instability in the
ligament (FIG. 21F). For the purpose of calculating the plateau
load, the positive steady state moment, which has been previously
developed in and, is considered. Here, the plateau load of the
ligaments is felt by the unit cell after each of the ligaments
exhibit snap-through. The positive steady state moment has the
following form:
M*.sub.+=(1+v)D.theta. (21)
Where D is a variable representing the following:
D = E .times. t 3 1 .times. 2 .times. ( 1 - v 2 ) ( 22 )
##EQU00012##
In Eqn 22, E is the elastic modulus and v is the poissons ratio of
the ligament base material (steel). The plateau forces
corresponding to the positive steady state moment can be obtained
with the positive steady state moment (Eqn 21) of the cylindrical
shell ligament. Thus, the axial load applied to each of the
ligaments individually post bending is given as the following
according to:
P plat = M + * [ ( L / 2 ) - .PHI. a .times. R ] .times. Sin
.function. ( ( .PHI. ) a ) + R ( 1 - Cos .function. ( .PHI. a ) (
23 ) ##EQU00013##
Thus, the total plateau load of the h-PXCM unit cell was formulated
using the general expression for the load felt by the unit cell
given in Eqn 16. Here F.sub.lig in Eqn 17 was replaced with Eqn
23.
F.sub.plat=P.sub.plat(-cos(.beta.+.theta.')+cos(60.degree.-.beta..degree-
..theta.')+sin(.beta.-30.degree.+.theta.')) (24)
To obtain more accurate expressions for the peak load and the
plateau load, more work is required to understand how the h-PXCM
rotates as a global compressive load is applied in addition to an
analysis of the system which considers the beams to bend
individually and not simultaneously.
Analysis
[0130] The design space chosen for the parametric analysis was
contoured with the energy dissipated per unit volume and the
average plateau stresses extracted from each of the simulations and
can be observed below in FIGS. 25A, 25C. Each of the points on the
contour plots represents an unit cell FE model that was used in the
analysis. The green point represents the FE model shown in FIG. 1b.
From the contour plots, it was observed that in the case that the
curvature is held static and the mass is changed, the slenderness
ratio, .pi..sub.1 controlled the plateau load that is experienced
by the unit cell and the ratio of circular support, .pi..sub.2
controlled the energy dissipated by the unit cell (FIGS. 25B, 25D).
The plots shown in FIGS. 25B, 25D show how the average plateau
stress as well as the energy dissipated per unit volume are
affected by .pi..sub.1 and .pi..sub.2 respectively.
[0131] The analytical equations derived in the previous section
represent our expectations of the behavior of the h-PXCM in the
scenario that both assumptions 1 and 2 are satisfied. FIGS. 26A,
26C display how the analytical equations (Eqns 19,23) predict the
peak and plateau loads to be exhibited by an h-PXCM unit cell for a
particular design based on the values of the dimensionless
parameters .pi..sub.1 and .pi..sub.2. The values predicted by the
analytical equations are compared with the results obtained from
the unit cell parametric analysis (FIGS. 26B, 26D). The analytical
equation for the peak load (F.sub.crit) predicted a decreasing
trend in the peak load required to induce snap-through as a
function of .pi..sub.1 similar to the peak load results obtained
from the parametric analysis (FIG. 26D). A similar trend was found
to be true in comparing the predictions of the analytical equation
for the plateau load against the results from the parametric
analysis. Note that the equation we use for the plateau load does
not consider the rotations applied to the cylindrical supports upon
loading and that we used the same angles for the peak load
equation. Additionally, the equations developed and utilized for
this analysis were not equipped to handle systems in which the
ligaments bend at different times. This is why the peak loads
estimated by the Eqn. 19 are so much larger than those from the
parametric analysis (FIGS. 26A-26B).
[0132] The energy dissipated per unit volume was plotted against
the average plateau stress for each h-PXCM unit cell FE model on an
Ashby Plot. The region housed by the h-PXCM, and regions occupied
by other cellular materials, are compared and shown below in FIG.
27. Note that only one material was used to model the cylindrical
shells and the cylindrical supports. If more than one material had
been used to model the h-PXCM unit cell, we expect that the
distribution of points on the Ashby plot would have been broader.
Note that the h-PXCM unit cell modeled with steel cylindrical shell
ligaments and rigid cylindrical supports was able to dissipate
relatively large amounts of energy for relatively low plateau
stresses as compared to other materials such as the sinusoidal PXCM
and the microlattices. The region of the Ashby plot occupied by the
h-PXCM suggests that the material could prove beneficial for a
myriad of biomedical applications as diverse as shoe supports that
enhance comfort and protect against back pain. Other applications
include countermeasures that protect against pedestrian cranial
injuries in vehicles, and headgear for bicyclists, those who play
contact sports, and the military.
[0133] We introduced the hexachiral phase-transforming cellular
material (h-PXCM) that utilizes the snap-through instabilities of
elastically deforming cylindrical shell ligaments to dissipate
energy for any in-plane loading direction without any dependence on
size effects under quasi-static conditions. The h-PXCM can be
designed with the use of two dimensionless parameters, the
slenderness ratio and the ratio of circular support, .pi..sub.1 and
.pi..sub.2 respectively. The parametric analysis of the h-PXCM unit
cell, which held the curvature of the ligaments constant and
changed the mass of the system, revealed the following about the
dimensionless parameters: (1) the slenderness ratio, .pi..sub.1
controlled the average plateau load of the unit cell model and (2)
the ratio of circular support, .pi..sub.2 controlled the energy
dissipated by the unit cell model. Additionally, the parametric
analysis revealed that an h-PXCM made of steel cylindrical
ligaments and rigid circular supports, will exhibit relatively high
energy dissipation for a relatively low plateau stress as compared
to other architectured materials, occupying an unfilled position on
the Ashby Plot.
[0134] Equations for the peak and plateau loads were developed from
a free body diagram of the h-PXCM unit cell system while
considering the following two assumptions: (1) That each
cylindrical shell in the h-PXCM unit cell is assumed to bend due to
compressive loads applied along the axis to the ligament length
before the moments take effect and (2) that bending occurs
simultaneously in each of the cylindrical shell ligaments. The
equations for the peak and plateau loads were observed to predict
similar trends compared to the results obtained from the unit cell
FE models. An extension of PXCM's with three dimensional structures
will be relegated to future work.
[0135] One embodiment of the present novel technology is
illustrate(in FIG. 28A-28B, an automobile or vehicle tire made of
concentrically layered bands or strips of bistable cells. cells are
operationally connected to define belts or layers, and the layers
are operationally connected in a concentric orientation to define
tires.
[0136] FIG. 29 illustrates another embodiment of the present novel
technology, a plurality of bistable cells configured to define an
energy absorbing and redistributing `earthquake resistant`
structural member.
[0137] While the novel technology has been illustrated and
described in detail in the drawings and foregoing description, the
same is to be considered as illustrative and not restrictive in
character. It is understood that the embodiments have been shown
and described in the foregoing specification in satisfaction of the
best mode and enablement requirements. It is understood that one of
ordinary skill in the art could readily make a nigh-infinite number
of insubstantial changes and modifications to the above-described
embodiments and that it would be impractical to attempt to describe
all such embodiment variations in the present specification.
Accordingly, it is understood that all changes and modifications
that come within the spirit of the novel technology are desired to
be protected.
* * * * *